Order Values 0E0Fb0

1. **State the problem:** We need to order the following values from least to greatest: $$\sum_{i=3}^8 i, 3\pi, 6, \sqrt{14}, 7!, \int_1^5 x\,dx, 8, 14, \infty, e^3, \log_3(13)$$ 2. **Calculate each value:** - $$\sum_{i=3}^8 i = 3 + 4 + 5 + 6 + 7 + 8 = 33$$ - $$3\pi \approx 3 \times 3.1416 = 9.4248$$ - $$6$$ (given) - $$\sqrt{14} \approx 3.7417$$ - $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$$ - $$\int_1^5 x\,dx = \left[ \frac{x^2}{2} \right]_1^5 = \frac{5^2}{2} - \frac{1^2}{2} = \frac{25}{2} - \frac{1}{2} = \frac{24}{2} = 12$$ - $$8$$ (given) - $$14$$ (given) - $$\infty$$ (infinity, largest) - $$e^3 \approx 2.7183^3 \approx 20.0855$$ - $$\log_3(13) = \frac{\ln 13}{\ln 3} \approx \frac{2.5649}{1.0986} \approx 2.334$$ 3. **List all values numerically:** $$\log_3(13) \approx 2.334$$ $$\sqrt{14} \approx 3.7417$$ $$6$$ $$8$$ $$3\pi \approx 9.4248$$ $$\int_1^5 x\,dx = 12$$ $$14$$ $$\sum_{i=3}^8 i = 33$$ $$e^3 \approx 20.0855$$ $$7! = 5040$$ $$\infty$$ 4. **Order from least to greatest:** $$\log_3(13), \sqrt{14}, 6, 8, 3\pi, \int_1^5 x\,dx, 14, \sum_{i=3}^8 i, e^3, 7!, \infty$$